Feynman Path Integral Formulation

7.5 Gravitational Wilson Loop 257Integration over the R’s results ink414 ATr(U2 + ε 2 I 4 )=− k 8 A (1 − 2 ε2 ) , (7.114)and the integration over the U’s (a sum over all possible orientations of the loop) istrivial.Next consider a less trivial situation where the Wilson loop goes around a numberof hinges and there is at least one internal hinge, i.e. a hinge where the elementaryloop surrounding it is not part of the Wilson loop. For simplicity, we shall considerthe case of one such loop. For the labeling of the rotation matrices and the hinges,the reader can annotate Fig. (7.11) in a way consistent with the expressions below.Fig. 7.11 A larger paralleltransport loop with twelveoriented links on the boundary.As before, the paralleltransport matrices along thelinks appear in pairs and aresequentially integrated overusing the uniform measure.The new ingredient in thisconfiguration is an elementaryloop at the center not touchingthe boundary.In this case, the lowest order contribution comes from a ninth-order term in theexpansion of the exponential of the action. One obtains the following result( ) ( )k 9 1 9 94 9 4∏ 17 A i Tr[(U C + εI 4 )(U 1 + εI 4 )] ∏ Tr(U i + εI 4 )i=1i=2( )9= k94∏ 18 A i ε 8 [Tr(U C U 1 )+4ε 2 ] . (7.115)i=1In this last equation one sets U 1 = U C +δU 1 , with < δU 1 >= 0, after which the sumover the loop’s orientation also becomes trivial. The above result also shows that itis better to take ε > 0, otherwise the answer vanishes to this order. But this is not aproblem, as the correct lattice action is recovered irrespective of the value of ε, asshown earlier in Eq. (7.101).Finally the value of a Wilson loop, when the loop is very large and surrounds nhinges, can be seen to be of the general form